Poor man's scaling: anisotropic Kondo and Coqblin--Schrieffer models

We discuss Kondo effect for a general model, describing a quantum impurity with degenerate energy levels, interacting with a gas of itinerant electrons, and derive scaling equation to the second order for such a model. We show how the scaling equation for the spin-anisotropic Kondo model with the power law density of states (DOS) for itinerant electrons follows from the general scaling equation. We introduce the anisotropic Coqblin--Schrieffer model, apply the general method to derive scaling equation for that model for the power law DOS, and integrate the derived equation analytically.


I. INTRODUCTION
Observed under appropriate conditions logarithmic increase (with the decreasing of temperature) of the scattering of the itinerant electrons by an isolated magnetic impurity was explained in 1964 in a seminal paper by Kondo, entitled "Resistance Minimum in Dilute Magnetic Alloy" 1 . Soon after it became clear that the phenomenon is manifested not only in resistivity, but in nearly all thermodynamic and kinetic properties 2 . The theoretical analysis of this effect turned out to be very fruitful, and led to the appearance of many approaches and techniques, which became paradigms in many totally different fields of physics. One of such approaches was the so called poor man's scaling, pioneered by Anderson 3 .
Though initially only magnetic impurity scattering was considered, later it became understood that similar effect can appear in case of a general quantum impurity. Such a general model was thoroughly reviewed in 1998 by Cox & Zawadovski 4 .
Recently the models where the density of states (DOS) of itinerant electrons in the vicinity of the Fermi level is the power function of energy has attracted a lot of interest [5][6][7][8][9][10][11] . The scaling was generalized to be applicable to such systems in the work by Withoff and Fradkin 12 .
Following the long line of works where spin-anisotropic Kondo model was studied 3,10,13,14 , we decided to revisit the problem of scaling equation for the Kondo effect in general 4 , and introduce and study the anisotropic Coqblin-Schrieffer (CS) model 15 in particular. Notice that it is known that the spin anisotropy can substantially change the physics of the Kondo effect in comparison with the isotropic case 4, [16][17][18] . The CS model, though being well studied previously 2,19-24 , draw a lot of attention recently in connection with the studies of quantum dots 25 , heavy fermions 26 and ultra-cold gases 27,28 .
The rest of the paper is constructed as follows. We formulate in Section II a poor man's scaling equation to second order for a general model, describing a quantum impurity embedded into a gas of itinerant electrons. We also generalize the equation to the case of power law be-havior of the DOS in the vicinity of the Fermi level. In Section III we show how the obtained earlier scaling equation for the spin-anisotropic Kondo model follow from the general scaling equation. In Section IV the XXZ CS model is introduced and scaling equation for this model is derived. Then everything is generalized to the case of the anisotropic CS model. The scaling equation both in the particular case of the XXZ CS model and in the general case of the anisotropic CS model are integrated analytically. We conclude in Section V. Some mathematical spin-offs are presented in the Appendix.

A. Kondo effect as explained by Kondo
The Hamiltonian we start from is 4 where c † kα and c kα are electron creation and annihilation operators of itinerant electron with wave vector k and internal quantum number α, ǫ k is the energy of the electron; X ba = |b >< a|, where |a >, |b > are the internal states of the scattering system, is the Hubbard X-operator.
In this paper we use the old-fashioned on-the-energyshell perturbation theory 29 following the paper by Kondo; similar approach was applied to the Anderson model by Haldane 30 . For the Hamiltonian (1) the transition probability per unit time from the initial state of the whole system m to the final state n is given to the second Born approximation by 1 where is the scattering matrix. Explicitly Eq. (3) takes the form 1 Writing Eq. (4) we assumed that in the state m all the electron states below the Fermi surface (corresponding to ǫ k = 0) are occupied, and there is an additional electron with the wave vector k and internal quantum number α; the scattering system is in the state |a >. In the state n all the electron states below the Fermi surface are again occupied, and there is an additional electron with the wave vector k ′ (ǫ k ′ = ǫ k = ǫ) and internal quantum number β; the scattering system is in the state |b >. In the R.H.S. of Eq. (4) the second term describes the processes when the electron with kα is first scattered to the unoccupied state qγ and then to k ′ β, and the third term describes the processes when an electron from an occupied state qγ is first scattered to k ′ β and then the electron with kα fills up the state qγ which is now empty 1 .
Equation (4) clearly explains the connection between the dynamics of the scattering system and the Kondo effect. For static impurity, Eq. (3) takes the form Because all the integrals with respect to energy in perturbation series terms are understood in the Principal Value sense, the denominator going to zero is by itself not a problem, and the second order terms just gives a correction to the matrix element of the order of the ratio of the scattering energy V to the band width (we consider electron band ǫ ∈ [−D 0 , D 0 ] and assume that the ratio is V /D is small). In addition, this correction is only weakly ǫ-dependent. On the other hand, each of the second order terms in Eq. (4) contains large logarithmic multiplier ln(ǫ/D 0 ), because of strongly asymmetric range of integration 2 ; due to the existence of the impurity quantum numbers, these terms do not add up to Eq. (5).

B. Scaling equation to second order
Equation (4), as it is written down, allows to calculate scattering at high temperatures. However, it allows more -to obtain a scaling equation for the Kondo model in the framework of the approach pioneered by Anderson 2,3 , which allows to selectively sum up the infinite perturbation series, using explicitly only the first two terms of such an expansion, as presented in Eq. (3).
We are interested only in the matrix elements between the electron states at a distance from the Fermi energy much less than the band width. The brilliant idea of Anderson, applied to the present situation, consists in reducing the band width of the itinerant electrons from [−D 0 , D 0 ] to [−D 0 − dD, D 0 + dD] (dD < 0) and taking into account the terms which corresponded to summation in Eq. (4) with respect to the electron states in energy intervals [−D 0 , −D 0 − dD] and [D 0 + dD, D 0 ] by renormalizing V βα,ba . Notice that such renormalization can be performed provided that |ǫ| ≪ D 0 and hence can be discarded in the denominators of the second order terms in Eq. (4). In our particular case, like in general, renormalization is the reduction of the Hilbert space H and changing the Hamiltonian H so as to keep the physical observable T constant. Thus we obtain where ρ is the density of states of itinerant electrons (assumed to be constant). Poor man's scaling consists in changing Eq. (6) to where D is now a running parameter. From Eq. (7) we obtain the scaling equation where Λ = D/D 0 (actually, the change of (implied) argument of each matrix element D → Λ in Eq. (8) was done for no reason), and, of course, Eq. (1) should be now understood as Let the matrix V βα,ba is presented as a sum of direct products of matrices, acting in ab and αβ spaces respectively where the set of matrices {G p } is closed with respect to commutation and hence generates some Lie algebra g; so is the set of matrices {Γ p } (Lie algebra γ).
With the help of Eq. (10) we can write down Eq. (8) in a more transparent form Introducing structure constants of the Lie algebras g and γ as f p st and ϕ π στ we can write down Eq. (8) in an even more transparent form Typically, the {G p } ({Γ π }) appear as infinitesimal operators of some Lie group G (Γ), and hence are Hermitian. Actually, this can be said the other way round. Assuming that the matrices {G p } ({Γ π }) are Hermitian, we see that the algebra g (γ) is real, and, hence, by Lie's third theorem is the Lie algebra of some simply connected Lie group 31 Consider an important particular case when γ ≡ g. (Matrices {G p } and {Γ π } not necessarily realize the same representation of the algebra, but have the same commutation relations; to emphasize that we will designate {Γ π } as {Γ p }.) If we assume that the matrix c pπ , in addition to being real, is symmetric, it can be diagonalized by a unitary transformation of the generators (such transformation does not change the commutation relations), and the matrix keeps its diagonal form in the process of renormalization. Thus Eq. (10) can be "reduced to the principal axes", that is to the form and Eq. (13) takes the form Equation (15) will be solved in Section III. General analysis of the equation for possible three-dimensional Lie algebras will be presented in Appendix A.

C. Power law DOS
The results of the previous Section can be easily generalized to the case when the electron dispersion law determines the power law dependence of the DOS upon the energy where r can be either positive or negative 12 . (We consider in this paper only particle-hole symmetric DOS; the influence of high particle-hole asymmetry on Kondo effect was studied in Ref. 32.) Notice that r = 0 corresponds to the previously considered case of flat DOS. In this case instead of Eq. (8) we obtain where G = CD r 0 is the DOS at the original band edges. Eq. (17)  Previously in this Section and in Section II we studied how the effective perturbation at a given energy changes, when the cut-off changes? Alternatively, we can ask ourselves: How does the effective perturbation at a given cut-off change, when the energy changes? That is we are interested in (dǫ can be of any sign). Looking at Eq. (4) we understand that

III. THE SPIN-ANISOTROPIC KONDO MODEL
To see how the general scaling equation is applied let us consider the following spin-anisotropic model (summation with respect to any repeated Cartesian index is implied) where S x , S y , S z are the impurity spin operators, σ x , σ y , σ z are the Pauli matrices, and J ij is the anisotropic exchange coupling matrix. The Hamiltonian (21) appears under many different names. Our opinion is that if we want to choose an eponymic name, the model should be called after T. Kasuya 34 . However, in line with the tradition we keep the name Kondo model, given because of important contribution to the derivation and analysis of the model made by J. Kondo 35 .
Taking into account the commutation relations where ǫ is Levi-Civita symbol, from Eq. (18) we obtain the scaling equation 10 (In this Section we measure J andJ in units of r/2G.) We assume that the microscopic tensor J ik entering into the Hamiltonian (21) is symmetric (no spin-orbit interaction 36 ). Analysis of the Hamiltonian in the presence of spin-orbit interaction see in Appendix B. In this case the tensor can be reduced to principal axes by rotation of the coordinate system, and it keeps it's diagonal form in the process of renormalization. So we can write down the interaction in a diagonal in cartesian indices (though non explicitly rotation covariant) form The scaling equation for this interaction is 10 where i, j, k are all different. The general solution of Eq. (25) (and, hence, of scaling equation) is written in terms of elliptic functions 10 where {α, β, γ} is an arbitrary permutation of {x, y, z}. Using the language of geometry, we say that each flow line lies on the surface of a special cone

A. The CS model
The CS model 15 is represented by the Hamiltonian where quantum number m changes from 1 to N . (To avoid cluttering, we omit in the equations in this Section the wave vector indices.) We changed sign of the constant J with respect to the original paper 15 For N = 2 the model coincides with the spin-isotropic Kondo model.

B. The XXZ CS model
If we demand that interaction (24) has SU (2) symmetry, it takes the form If we reduce the symmetry to U (1), interaction (24) takes the form we will call such exchange interaction the XXZ Kondo model. Equation (32) can be written down using Hubbard operators Motivated by Eq. (33) we suggest the following Hamiltonian for arbitrary N , which we for obvious reasons will call the XXZ CS model, (An alternative motivation for introducing the model can be found in Appendix C.) Further on in this Section we'll present the calculations only for the case of constant DOS, and only the final results will be written down for the case of the power law DOS. For the interaction (34) scaling equation (11) becomes For the case of isotropic CS model and for the case N = 2, Eq. (35) is reduced to the well established results 2 .

C. Integration of the scaling equation for the XXZ CS model
Dividing two equations in (35) by each other we obtain homogeneous differential equations of the first degree, which can be easily integrated where C is an arbitrary constant. Substituting the solution (36), (37) into Eq. (35) we obtain Stable fixed point correspond to phases of Eq. (35), characterized by the asymptotic behaviour of the flow lines. The part of the phase plane J x > −(2/N )J z , 0 is characterized by the attractor J x = J z > 0, and will be called the Kondo phase I. Notice that in this phase, the SU(N ) symmetry, which is absent for the microscopic Hamiltonian, is being recovered in the process of scaling. The part J x < −J z , 0 is characterized by the attractor J x = −(2/N )J z < 0, and will be called the Kondo phase II. In the part −(2/N )J z ≥ J x ≥ J z the flow lines are attracted to the fixed points at the axis J x = 0.
A flow line can reach the fixed point w = 0 only in the end of infinitely long evolution, which is a common situation for a fixed point. Hence Ising model is obtained only in the infrared limit. However, in a Kondo phase a flow line reaches fixed point w = 1 or w = −2/N after finite evolution (the R.H.S. of Eq. (38) being non-analytic function of w at these points). Singularity of J x , J z at finite value of Λ is another indication of the limited applicability (in the Kondo phases) of the truncated to second order scaling equation.
To the general solution (36) we should add singular Once again we see that evolution starting in one of the Kondo phases hits the singular point at finite value of ln Λ. Equation (36) allows us to plot the flow diagram of the scaling equation, which is presented on Fig. 1. For the sake of definiteness we have chosen N = 4. Notice that the flow diagram is qualitatively similar to that of the XXZ Kondo model 2 . For the case of the power law DOS Eqs. (36) and (38) and (Eq. (39) is changed similarly).
Eqs. (40) and (41) is a rigorous but a bit formal result. In particular, it demands some effort to extract out of them the fixed points of the original scaling equation which can be easily found by inspection of Eq. (42). The equation has a trivial fixed point (J * x , J * z ) = (0, 0), which is stable for r > 0 and unstable for r < 0, and two semistable (critical) non-trivial fixed points

D. The anisotropic CS model
To formulate the general anisotropic CS model let us return to the spin-anisotropic Kondo model. The interaction can be written down using Hubbard operators Motivated by Eq. (44) we suggest the following interaction for arbitrary N For this interaction scaling equation (11) becomes Like in the previous Subsection we obtain after inte- Analog of Eq. (38) can be presented as Thus in the phase space with the coordinates J x , J y , J z there are 4 Kondo phases, each defined by the attractor of all the flow lines, given by one of the vectors: In addition there are 3 Ising phases, each defined by the fixed points of all the flow lines, lying on one of the lines: J x = J y = 0, J x = J z = 0, J y = J z = 0. Further analysis of the phase diagram we postpone until later.
For the case of the power law DOS Eqs. (47) and (49) become and In the end, notice that it would be interesting to apply the approach presented in this paper to the case, where the exchange is influenced by Rashba 38 or Dzyaloshinsky-Morya-Kondo interaction 36 , to the case when the DOS has a logarithmic singularity 39 , to the multi-channel Kondo model 40 , and also to the problem of competition between the Kondo effect and RKKY interaction 41 . Another possibly interesting field for application of the presented approach is the models when orbital and spin degrees of freedom of the impurity co-exist 42 .

V. CONCLUSIONS
In the present contribution we derive the poor man's scaling equation to the second order for a general model, describing a quantum impurity with degenerate energy levels embedded into a gas of itinerant electrons, both for flat and for the power law DOS. We show how the obtained previously scaling equations for spin-anisotropic Kondo model follow from the general scaling equation.
We introduce the anisotropic CS model, and the XXZ CS model as its particular case. We apply the general scaling equation to derive scaling equations for these models. We integrate analytically the scaling equation both in the particular case of the XXZ CS model and in the general case of the anisotropic CS model.